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Math Help - taylor's theorem

  1. #1
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    taylor's theorem

    Use Taylor's Theorem to prove that
    cosx = \sum_{k=0}^{\infty} \frac{(-1)^k}{(2k)!} x^{2k}

    Thanks!
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  2. #2
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    The Taylor series for the function f(x), centred at a = 0, is given by: f(0) + f'(0)x + \frac{f''(0)}{2!}x^2 + \cdots + \frac{f^{(n)}(0)}{n!}x^n + R_n(x)

    where R_n(x) = \frac{f^{(n+1)}(c)}{(n+1)!}x^{n+1} is the remainder term ( c \in (0,x)). To prove that the Taylor series of f(x) converges to f(x) for all x, we must show that the interval of convergence is \infty and \lim_{n \to \infty} R_n (x) = 0.

    Coefficients will be for you to find. Use the ratio test to find the interval of convergence.

    For the remainder term, notice that: 0 < |R_n (x)| = \left| \frac{\cos^{(n+1)} (c)}{(n+1)!}x^{n+1}\right| {\color{red}\ < \ } \frac{x^{n+1}}{(n+1)!}

    Use squeeze theorem to finish off (recall: R_n (x) \rightarrow 0 iff  |R_n (x)| \rightarrow 0))
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